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The Schroder and Konig iteration schemes to find the zeros of a (polynomial) functiong(z) represent generalizations of Newton's method. In both schemes, iteration functionsf m (z) are constructed so that sequencesz n+1 =f m (z n ) converge locally to a rootz * ofg(z) asO(|z n −z *|m). It is well known that attractive cycles, other than the zerosz *, may exist for Newton's method (m=2). Asm increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The Konig functionsK m (z) appear to minimize such perturbations. In the case of two roots, e.g.g(z)=z 2−1, Cayley's classical result for the basins of attraction of Newton's method is extended for allK m (z). The existence of chaotic {z n } sequences is also demonstrated for these iteration methods.

Let B B be the Julia set associated with the polynomial T z = z N + k 1 z N − 1 + ⋯ + k N Tz = {z^N} + {k_1}{z^{N - 1}} + \cdots + {k_N} , and let μ \mu be the balanced T T -invariant measure on B B . Assuming B B is totally real, we give relations among the entries in the infinite-dimensional Jacobi matrix J J whose spectral measure is μ \mu . The specific example T z = z 3 − λ z Tz = {z^3} - \lambda z is given, and some of the asymptotic properties of the entries in J J are presented.

It is argued that since any real space renormalisation R is a transformation under which the partition function is invariant, those R which are based upon various real space approximation schemes can in principle be extended into the complex field of the parameter space. In this extension the Julia set of R (now extended to include R which are algebraic) is an approximation to the locus of the Yang-Lee zeros. Finite-size scaling using Onsager's solution of the two-dimensional Ising model is used to construct R which are highly accurate in real temperatures in the critical region; these are extended into the complex temperature plane and results are displayed for both the anisotropic and isotropic cases.

We calculate the fractal dimension D (t) of the Julia set associated with the partition function zeros of the ferromagnetic Ising model on the Cayley tree. In the low-temperature phase the Julia set is the unit circle (D (t) = 1) in the complex fugacity plane. In the paramagnetic phase the Julia set is a Fatou dust (Cantor set) on the unit circle whose fractal dimension is a decreasing function of the temperature. Like an order parameter D (t) displays a singular behaviour with a critical exponent βD = 1/2.

In this paper the orthogonality properties of iterated polynomials are shown to remain valid in some cases for rational maps. Using a functional equation fulfilled by the generating function, the author shows that the Hausdorff dimension of the maximal measure is a real analytical function of the coefficients of an Axiom A rational map satisfying the property that all poles of f and zeros of f'(z) have multiplicity one. Here we will consider f a rational map such that the Julia set (see [1]) is bounded and f is of the form f(z) = P(z)(Q(z))-', where P(z) = zn + ani zn+ +a1z + ao, Q(z) = bdzd + bdlzd-i + +b z + bo, where ai E C, bj E C, 1bd =O, n > 2, and d < n. In [6, 8, and 10] it was shown that for f a rational map there exists just one f-invariant probability measure u such that, for any continuous function 4), 4)(x) du(x) = n-' E ?(zi(x)) du(x), where zi(x), i E {1, .. ., n }, are the roots of f(z) = x, counted with multiplicity, and this is the measure of maximum entropy. This measure is called the maximal measure, and it has entropy log n. For f such that f(ox) = cx and J(f) bounded, this measure is the equilibrium measure for the logarithm potential if and only if f is a polynomial [1, 9]. Let F(z) be the only one such that F(z)/z is analytic near x, F(z) z as z -b o, and F'(z)F(z)i = J (z x)-ldu(x) = z-1( E MmZ -), where Mm = Jxm du(x) for m E N (see [2]) are the m-moments of u. Note that Mo = 1, and the expansion is valid only when the Julia set is bounded, which implies either d < n 1 or d = n 1, and I bdI < 1 or IbdI = 1, and there is a Siegel disk around infinity. Received by the editors August 16, 1985.

For realλ a correspondence is made between the Julia setBλ forz→(z−λ)2, in the hyperbolic case, and the set ofλ-chainsλ±√(λ±√(λ±..., with the aid of Cremer's theorem. It is shown how a number of features ofBλ can be understood in terms ofλ-chains. The structure ofBλ is determined by certain equivalence classes ofλ-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics ofλ-chains. The functional equations obeyed by attractive cycles are investigated, and their relation toλ-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets andλ-chains. Certain “Julia sets” associated with the Feigenbaum function and some theorems of Lanford are discussed.

Let $f:\hat{C}\to\hat{C}$ be a subhyperbolic rational map of degree $d$. We construct a set of coding maps $Cod(f)=\{\pi_r:\Sigma\to J\}_r$ of the Julia set $J$ by geometric coding trees, where the parameter $r$ ranges over mappings from a certain tree to the Riemann sphere. Using the universal covering space $\phi:\tilde S\to S$ for the corresponding orbifold, we lift the inverse of $f$ to an iterated function system $I=(g_i)_{i=1,2,...,d}$. For the purpose of studying the structure of $Cod(f)$, we generalize Kenyon and Lagarias-Wang's results : If the attractor $K$ of $I$ has positive measure, then $K$ tiles $\phi^{-1}(J)$, and the multiplicity of $\pi_r$ is well-defined. Moreover, we see that the equivalence relation induced by $\pi_r$ is described by a finite directed graph, and give a necessary and sufficient condition for two coding maps $\pi_r$ and $\pi_{r'}$ to be equal., Comment: 27 pages, 5 figures

Independence polynomials of graphs enjoy the property of essentially being closed under graph composition (or ‘lexicographic product’). We ask here: for higher products of a graph G with itself, where are the roots of their independence polynomials approaching? We prove that in fact they converge (in the Hausdorff topology) to the Julia set of the independence polynomial of G, thereby associating with G a fractal. The question arises as to when these fractals are connected, and for graphs with independence number 2 we exploit the Mandelbröt set to answer the question completely.

Our goal in this paper is to describe some of the interesting topology that arises in the dynamics of entire functions such as the complex exponential family Eλ(z)= λe. We will see that the important invariant sets for this family possesses a extremely rich topological structure, including such objects as Cantor bouquets, Knaster continua, hair transplants, and explosion points. For a complex analytic function E, the interesting orbits lie in the Julia set, which we denote by J (E). This is the set on which the map is chaotic. For the exponential family, the Julia set of Eλ has three characterizations: (1) J (Eλ) is the set of points at which the family of iterates of Eλ, {En λ} is not a normal family in the sense of Montel. This is the characterization that is most useful to prove theorems. (2) J (Eλ) is the closure of the set of repelling periodic points of Eλ. This is the dynamical definition of the Julia set. (3) J (Eλ) is the closure of the set of points whose orbits tend to ∞. This is the characterization that is most useful to compute the Julia set. We remark that characterization 3 differs from the case of polynomial iterations, where the Julia set is the boundary of the set of escaping orbits. The reason for the difference is Eλ has an essential singularity at ∞, while polynomials have superattracting fixed points at ∞. The equivalence of (1) and (2) was shown by Baker, see [6]. The equivalence of (1) and (3) is shown in [19]. In this paper we will concentrate on the dynamics of Eλ where λ is real. For λ positive, the Julia set for Eλ undergoes a remarkable transformation as λ passes through 1/e. We will show below that Eλ possesses an attracting fixed point when 0< λ< 1/e. All points in the left half plane have orbits that tend to this fixed point. Indeed, the full basin of attraction of this fixed point is open and dense in the plane.

Let M denote the class of functions f meromorphic outside some compact totally disconnected set E = E ( f ) and the cluster set of f at any a ∈ E with respect to E c = C ˆ \ E is equal to C ˆ . It is known that class M is closed under composition. Let f and g be two functions in class M, we study relationship between dynamics of f ○ g and g ○ f . Denote by F ( f ) and J ( f ) the Fatou and Julia sets of f. Let U be a component of F ( f ○ g ) and V be a component of F ( g ○ f ) which contains g ( U ) . We show that under certain conditions U is a wandering domain if and only if V is a wandering domain; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U unless U is a Siegel disk or Herman ring.

We characterize the Julia sets of certain exponential functions. We show that the Julia sets J ( F λ n ) of F λ n ( z ) = λ n e z n where λ n > 0 is the whole plane C , provided that lim k → ∞ F k λ n (0) = ∞. In particular, this is true when λ n are real numbers such that λ n > 1 ne 1/ n . On the other hand, if 0 λ n 1 ne 1/ n , then J ( F λ n ) is nowhere dense in C and is the complement of the basin of attraction of the unique real attractive fixed point of F λ n . We then prove similar results for the functions[formula] where λ i ∈ C − {0}, 1 ≤ i ≤ n + 1, a j > 1, 1 ≤ j ≤ n , and m , n ≥ 1.

We consider the exponential maps ƒ λ : ℂ → ℂ defined by the formula ƒ λ ( z ) = λ e z , λ(0,1/ e ]. Let J r (ƒ λ ) be the subset of the Julia set consisting of points that do not escape to infinity under forward iterates of ƒ. Our main result is that the function λ h λ :=HD( J r (ƒ λ ),)), λ(0, 1/ e ], is continuons at the point 1/ e . As a preparation for this result we deal with the map ƒ 1 / e itself. We prove that the h 1 / e -dimensional Hausdorff measure of J r (ƒ 1 / e ) is positive and finite on each horizontal strip, and that the h 1 / e -dimensional packing measure of J r (ƒ λ ) is locally infinite at each point of J r (ƒ λ ). Our main technical devices are formed by the, associated with ƒ λ , maps F λ defined on some strip P of height 2π and also associated with them tonformal measures.

We examine inverse limits of postcritically finite polynomials restricted to their Julia sets. We define the “trunk” of a Julia set, a forward-invariant set related to the Hubbard tree, and use it to show that the inverse limit always contains at least one indecomposable subcontinuum. We characterize when the inverse limit is indecomposable and also examine how the trunk behaves in the inverse limit.

The Julia set of Ramanujan's iterated exponential map, Fr+1(z) = exp {Fr(z)} -1, is investigated for positive integral r. Connection with Devaney's work on the map zr+1 = λ exp[zr] is also made.

The self-replicating properties of the Julia sets Jc(p) for the iterative processes z implies zp+c are examined for integers p)1. It is shown that the corresponding Mandelbrot sets M(p) contain (p-1)-fold symmetries, which results in Jc(p) having p-fold symmetries.

Let {f c } be a family of mappings dependent on a parameter c. A method is presented for calculating sets which can represent the successive derivatives with respect to c of the Julia set for f c at some value c 0 of c. In practice, the Taylor series constructed from these derivatives yields good approximations to the Julia sets for f c for c in a neighborhood of c 0.

For polynomials of second order, p(x) = a2x2 + a1x + ao, an almost complete classification of the corresponding Julia sets can be given in terms of the Mandelbrot set. First note that p(x is conjugate to p c (z)=z2 + c by means of the coordinate transformation \( x \mapsto z = a_2 x + a_1 /2,with{\text{ }}c = a_0 a_2 + \frac{{a_1 }} {2}\left( {1 - \frac{{a_1 }} {2}} \right). \)

Recurrent iterated function systems generalize iterated function systems as introduced by Barnsley and Demko [BD] in that a Markov chain (typically with some zeros in the transition probability matrix) is used to drive a system of maps w j : K→ K, j = 1, 2,…, N, where K is a complete metric space. It is proved that under “average contractivity,” a convergence and ergodic theorem obtains, which extends the results of Barnsley and Elton [BE]. It is also proved that a Collage Theorem is true, which generalizes the main result of Barnsley et al. [BEHL] and which broadens the class of images which can be encoded using iterated map techniques. The theory of fractal interpolation functions [B] is extended, and the fractal dimensions for certain attractors is derived, extending the technique of Hardin and Massopust [HM]. Applications to Julia set theory and to the study of the boundary of IFS attractors are presented.

Agraïments: The authors wish to thank Christian Henriksen for the algorithm to compute nonsymmetric Herman rings of transcendental maps, as the one in Fig. 2. The authors are grateful to the EU network CODY for giving the opportunity to many young researchers like the second author to enjoy postdoctoral research stays as the one that made this paper possible. They also thank the referee for reading the paper and giving useful suggestions. We prove some results concerning the possible configurations of Herman rings for transcendental meromorphic functions. We show that one pole is enough to obtain cycles of Herman rings of arbitrary period and give a sufficient condition for a configuration to be realizable.

On considere l'ensemble de Julia pour l'application rationnelle complexe Z→Z 2 −λ+e/Z, ou λ et e sont des parametres complexes dans la limite e→0